evaluatiion of metamorphic conditions

CHAPTER 3

EVALUATION OF METAMORPHIC CONDITIONS

The nature of the most stable mineral assemblage in a rock of a given composition is aconsequence of the minimization of the free energy G of the system. As G is a function ofP and T, there is, therefore, a direct liaison between a paragenesis and its crystallizationconditions. The evaluation of these conditions is a difficult exercise which has aroused theinterest of petrologists for decades. The results which can presently be obtained are notcompletely satisfactory; however, without being able to determine the absolute values of Pand T with great precision, it is still possible to characterize metamorphic gradients as wellas their evolution in time conveniently from the data provided by the metamorphic paragen-eses. These parameters are the basis for thermal models which themselves reflect thegeodynamic mechanisms responsible for the formation of mountain chains.

An elementary method, which can be used in the field with the aid of a simple hand lens,allows (in principle) mapping ofsurfaces ofequal metamorphic intensity, or isograds. Theseisograds are based on the appearance, or eventually the disappearance, in the series, ofcharacteristic minerals or index minerals. These observations have demonstrated, sincethe initial work of Tilley in 1925, that successive zones are the result of conditions ofincreasing metamorphic grade. The chlorite, biotite, garnet, staurolite, kyanite, sillimanitezones, for example, appearin the metapelites of the Scottish Highlands or of the LowerLimousin (Fig. 29).

Mapping isograds is the fundamental field technique for the study of a metamorphicseries, but this method is too crude to allow a detailed interpretation. Three factors, inparticular, result in the index minerals becoming unreliable;1) the appearance or disappearance of a mineral phase does not depend uniquely on meta-

morphic conditions, but also on the composition of the rock. In heterogeneous seriesthe distribution of index minerals is of no use in mapping an isograd surface, especiallyif the metamorphic zones are oblique to the lithology.

2) The appearance or disappearance of a phase is the result of mineral reactions. Thedifferent reactions involved in the crystallization of an index mineral do not necessarilytake place under the same conditions. The appearance of kyanite couldresult from the following reactions:

pyrophyllite = kyanite + quartz + watermuscovite + quartz = kyanite + microcline + watergarnet + muscovite + quartz = kyanite + biotite + waterstaurolite + quartz = kyanite + garnet + water

Therefore, to map the appearance of kyanite without paying attention to the nature ofthe reactions which are responsible for the crystallization of the mineral provides littlereliable information on the real distribution of metamorphic isograds.3) The spatial distribution of index minerals depends on the deformation which affected

47

INDEX MINERALS AND METAMORPHIC ISOGRADS

48 CHAPTER 3

the series after crystallization. Isograds or pseudo-isograds which can be proven tohave been considerably deformed offer, as a result, no direct significance to the inter-pretation of metamorphism. Figure 30 shows the structural complexity of the LowerLimousin, which, after a second look, does not lend itself to a clear study of metamor-phic isograds, using only index minerals.

METAMORPHIC FACIES

Methods of evaluation of metamorphic intensity based on the study of metamorphic faciesor mineral facies were introduced by Eskola in 1915. They were further developed byTurner and Verhoogen in 1960 and popularized by Winkler from 1965 on.

3.1 Use of metamorphic facies

Mineral assemblages of rocks are identified by the polarizing microscope. They may becompared to lists of characteristic minerals, or, better, to diagrams prepared and refinedthrough usage. Lists of type minerals characterizing the different facies are furnished in theappendix.

Diagrams allow qualitative evaluation of metamorphic conditions. P-T space is di-vided into a certain number of areas (Fig. 31), each representing a metamorphic facies;each facies includes all metamorphic rocks; or, more precisely, all mineral assemblageswhich result from approximately the same P-T conditions, whatever their chemical compo-sition. Each metamorphic facies, therefore, contains all the possible mineral assemblagesstable under the conditions considered. The metamorphic facies carry conventional nameswhich should not be taken in the literal sense; the greenschist facies, for example corre-

Evaluation of metamorphic conditions 49

sponds to the P-T conditions under which rocks of a certain composition (metabasites)crystallized as a greenschist (epidote + actinolite ± chlorite + albite ± quartz assemblage).

The same diagrams allow an approximation of the overall chemical composition of therocks. From this estimate, hypotheses as to the nature of the protolith may be advanced.

3.1.1 Choice of typomorphic mineral assemblagesIt is critical to take into consideration those equilibrium assemblages resulting from thesame recrystallization period. Depending on the case, these typomorphic mineral assem-blages (so-called because they characterize a metamorphic facies) are syn- or post-kine-matic according to the importance of their development in the rocks under consideration.It is essential to compare any assemblage with one chronologically comparable within thesame metamorphic series.

3.1.2 Choice of a projection systemIn order to describe a metamorphic series in all its diversity better, and in order to make

50 CHAPTER 3

comparisons between different series, it is useful, even essential, to use a tool which canpresent a large variety of chemical compositions. A well-developed system for presentingthe principal metamorphic facies is the combination of two diagrams, the ACF and the

(Fig. 32; simplified construction described in the appendix) which allows character-ization of the majority of usual mineral assemblages. This system, popularized by H.G.F.Winkler, does, however, have several disadvantages:1) by its construction, it is impossible to show sodic phases; this is a major disadvantage

in the study of high pressure-low temperature metamorphism, in whose paragenesesalbite (sodic plagioclase), glaucophane (sodic amphibole) and jadeite (sodic pyroxene)are critical minerals.

2) it combines iron (Fe) and magnesium (Mg). These two cations have an ionic radiussimilar to one another (0.74 and 0.66 Å, respectively) and often play comparable rolesin the occupation of structural sites in minerals. In practice Fe and Mg are not distrib-uted in the same way between two or more ferromagnesian phases in equilibrium (gar-


net-biotite, for example or even orthopyroxene-clinopyroxene). It is therefore oftennecessary to use a projection which distinguishes between Fe and Mg, such as the

52 CHAPTER 3

AFM diagram of Thompson (Fig. 32a, construction described in appendix) in parallelwith diagrams.

3) it does not take into account assemblages unsaturated in (i.e. without quartz).Water is always considered in excess, a serious handicap for the metamorphic domainswhere the fluid phase is rich in (granulite facies).Despite these disadvantages, the diagram used in conjunction with the

AFM, is a vital tool for rapid analysis of a metamorphic series, except for rocks in the highpressure-low temperature series.

3.1.3 Use of triangular projections; interest and limitsThe point which represents a rock in a triangular diagram is enclosed in a smaller triangledefined by the three minerals constituting the type paragenesis ofthe rock. This simple ruledevolves from the application of the phase rule (cf. below). It is often difficult to applywithout a bit of thought, because the assemblages generally contain more than three phasesin equilibrium. It is therefore necessary to present the assemblage in several types of dia-grams simultaneously. For two different assemblages (Fig. 33):

(1) muscovite + biotite + K feldspar + quartz(2) muscovite + cordierite + staurolite + sillimanite + plagioclase + quartz

Assemblage (1) appears as a single point on the diagram, quartz being in excessby definition, the equilibrium point of a field of four phases is conveniently defined by one


projection point. No calcic minerals occur in the paragenesis, and there is not even a pointrepresenting this rock in the ACF diagram.

Assemblage (2) is more difficult to treat:it contains a calcic mineral (plagioclase) and a potassic mineral (muscovite); it should beprojected simultaneously in the ACF and diagrams,in each diagram it corresponds to two representative points;

2a muscovite + staurolite + cordierite + plagioclase + quartz2b muscovite + staurolite + sillimanite + plagioclase + quartz

In reality, these two points are only the same because this represents an equilibriumparagenesis. Therefore it is the projection system which is not entirely satisfactory. Thismay be verified in the AFM diagram where a paragenetic triangle (sil + st + crd) resultsfrom the fact that staurolite and cordierite in equilibrium do not have the same Fe/Mg ratio.The combination of these two elements makes the presentation of this triangle impossiblein the diagrams.

Therefore, the imperfect nature of these triangular projections must be accepted inorder that, in simplifying the composition of the system into three constituents, only arestricted number of phases may be considered, three only, when it is often necessary toconsider four or five. This aspect is approached in a more theoretical manner in the secondpart of this chapter. Despite these imperfections, the triangular projections allow simulta-neous:a) evaluation of the chemical components of a metamorphic rock on the basis of its min-

eral assemblage, or the approximation of its protolith.b) evaluation of the metamorphic conditions.

3.1.3.1 Evaluation of the chemical composition of a rock. The compositional domain ofdifferent sedimentary and igneous rocks are shown in the diagram; the position of the pointrepresenting a rock with respect to these fields serves as a guide for the definition of theprotolith (Fig 32b). Certain domains are, however, ambiguous because they characterizeboth the field of basalts (for example) as well as that of carbonate-rich pelites (marls).Supplementary observations are necessary to resolve these ambiguities; such as the abun-dance of Ti-rich phases (sphene, rutile) which characterize an igneous protolith.

3.1 3.2 Evaluation of metamorphic conditions. Metamorphic rocks are classified, as afunction of the nature of the mineral assemblage, into conventional pigeonholes whichrepresent the mineral facies in a P-T diagram (Fig. 31) elaborated from field observationsas well as theoretical and experimental data. A rapid microscope study allows immediateclassification of one assemblage in relation to another in the same metamorphic series; orbetween different metamorphic series. This handy, simple and rapid method has, neverthe-less, two serious flaws.

Semantic ambiguity in the naming of facies. The amphibolite facies groups all the assem-blages which crystallized at P-T conditions under which green hornblende and plagioclaseare produced in rocks of appropriate composition; however, rocks which do not containany hornblende also belong to the amphibolite facies (sil + bt + Kfs + qtz or crd + grt + bt+ qtz; etc.). Furthermore, certain amphibole-bearing assemblages do not belong to the

54 CHAPTER 3

amphibolite facies (actinolite + epidote + albite; glaucophane + lawsonite + albite; etc.) butto the greenschist or glaucophane schist facies respectively.

An approximate localization of mineral assemblages in the P-T diagram. The fields ofcertain metamorphic facies are extremely large (greenschist, amphibolite, granulite). Somuch so, that metamorphic rocks which crystallize under very different P-T conditionsfrom one another, are likely to be classified within the same metamorphic facies.

The first fault is rapidly overcome by learning the real significance of the terms. Thesecond is lessened by the introduction of subfacies which allow location of the assemblagesin P-T space with a greater precision.

3.2 Metamorphic facies and subfacies

The utility of metamorphic facies rests in the possibility of immediate characterization, inthin section, of the metamorphic conditions for a large range of composition, with respect


to a standard diagram (Fig. 34). It is for this reason that the conditions are poorly defined,or rather defined in a general fashion. In fact a facies takes into consideration a large num-ber of mineral assemblages governed by a number of mineral reactions dispersed in P-Tspace. A more precise localization ofan assemblage is possible by taking into account onlynarrow compositional domains, thus limiting the number of mineral reactions. This moreprecise, but less general, localization, brings up the definition of metamorphic subfacies. Inthis perspective, the petrologist’s ideal metamorphic series is one which presents a largevariety of rocks of different composition, closely associated to one another (an initial marl-pelite sequence, for example). The diversity of mineral assemblages will allow subdividingthe facies into a large number of subfacies, thanks to different mineral reactions whichbehave in a different fashion for rocks of differing composition.

Examples of subdivision into subfacies are given in Figure 35. In practice, once thetechnique ofdivision into subfacies is applied, there is no longer arigorously defined scheme,and everyone is able to suggest subfacies which are best adapted to the study of a particularseries.

56 CHAPTER 3

3.3 Grades or degrees of metamorphism, another way of presenting facies andsubfacies.

The evaluation of the intensity of metamorphism is based on certain key reactions whichsegment the field of increasing metamorphic intensity; very low grade, low grade, moder-ate grade and high grade (Fig. 36). These reactions apply to rocks of different composi-tion: pelitic and granitic; basaltic and andesitic; carbonate; etc. The method is similar tothat of mineral facies, but the limits of the fields are defined more rigorously, in closerelation to mineral reactions. It has the advantage of setting aside the facies nomenclaturewhich is too closely associated with particular rock compositions, for a more objectivenomenclature. However this cannot bring the same precision as that which may be drawnfrom a study of subfacies.

GEOMETRICAL ANALYSIS

Mineral reactions are the definitive elements in the petrologic analysis of metamorphicseries. They correspond to a particular case of minimization of the free energy of a system,in which the assemblage which has the lowest G value develops preferentially under theconditions considered. The slopes of these reactions are controlled by the difference involume and entropy between the reactants and the products. In a particular system thereactions are not independent of one another. Their arrangement in P-T space is closelycontrolled by the phase rule and geometrical constraints which follow from the principle ofminimization of free energy. The geometrical analysis (the rules for which were introducedby Schreinemakers, and developed by Zen) of a system allows elaboration of petrogeneticgrids, which are extremely valuable tools for the study of metamorphic series and consti-tute the basis for all experimental investigations. The concerted use of geometrical analysis


and thermodynamic data results in a more precise petrologic analysis of metamorphic con-ditions, even if the absolute values of P and T are not always known precisely.

3.4 Definitions and reminders

A rock is a complex system either open or closed. It consists of a certain number of phases,either solid (minerals), liquids (glasses) or fluids. In metamorphic rocks there is no liquidphase, except in the particular case of migmatites. In what follows in this chapter, onlyclosed systems are considered. In these topochemical systems the mass of chemical con-stituents does not change during reactions; both terms of an equilibrium equation are con-stituted of the same number of molecules of the same independent constituents.

The independent constituents are the chemical constituents necessary and sufficient todescribe all the phases of the system. This is a fundamental notion which depends on thereactions studied in the system. Thus, for the reaction:

(1) andalusite = sillimaniteOnly one independent constituent is necessary and sufficient to define the two

phases, andalusite and sillimanite which have the same composition. But the reaction:(2) andalusite = corundum + quartz

is described by two independent constituents The system is seeminglychemically analogous to that established for reaction (1).

The reaction:(3) quartz + calcite = wollastonite +

is described by three independent constituents; but a fourth constituentmust be taken into account in the case where a low oxygen fugacity results in the reaction:

(4)In practice what may be defined as an independent constituent of a system is all the

chemical elements present in sufficient quantity to control the stability of a specific phase.For example is not considered as an independent constituent when it is dissemi-nated in the crystal structure of biotite, amphibole or pyroxene. It becomes an independentconstituent if it is in sufficient quantity to allow the formation of a titaniferous phase such asrutile or sphene.

3.5 Variance of a system: the phase rule

The phase rule which establishes a relation between the variance of a system and thenumber of independent constituents n, the number of intensive variables v, and the numberof phases is written:

3.5.1 The idea of variance or degrees of freedom of a systemDivariant equilibria. Their stability fields in the diagram are the surfaces limited by tworeactions. These surfaces represent two degrees of freedom: in effect P and T may varyindependently, one from the other, within defined limits, without changing the nature of theassemblage. These surfaces are surfaces of divariant equilibrium for which = 2.

58 CHAPTER 3

Univariant equilibria. These are the lines or curves of the reactions themselves. Thesereactions have only one degree of freedom. They represent the many variations of anintensive parameter as a function of the other to maintain equilibrium. These equilibria areunivariant = 1.

Invariant equilibrium. Such an equilibrium is shown in the figure by the point of conver-gence of the three univariant curves. It is defined by one value for each of the intensiveparameters, and has no degree of freedom. It is an invariant point for which = 0.

In a system influenced by three intensive variables (P, T, for example), an assem-blage with three dimensions has three possible degrees of freedom, it is trivariantAs indicated by the phase rule, the number ofphases in equilibrium in a system depends onthe variance:

for

3.5.2 Number of assemblages possible in a system as a function of the varianceSuppose a system has n independent constituents. The total number of phases in equilib-


rium for the conditions of an invariant point is equal to n + v . How many possibili-ties C, are there to distribute these phases at:

invariant equilibrium for (n + v) phasesunivariant equilibrium for (n + v -1) phasesdivariant equilibrium for (n + v -2) phases

The reply is given by the combining formula which indicates the number of possibili-ties C to combine K objects taken m by m.

is the total number of phases available m is the number of phases inequilibrium for each value of the variance: In consequence:

becauseSuppose a system under two intensive variables (P and T): n + 2 phases can only be

combined at one invariant point, n. However, the calculation shows there are (n + 2)univariant equilibria. The number of possible divariant equilibria increases rapidly as afunction of n (Table 2).

3.6 Applications to simple systems

3.6.1 Systems with one independent constituentConsider the system already examined (Fig. 38a). The phase rule and the combin-ing formula show that three phases are in equilibrium at one invariant point when twointensive variables are imposed on the system. The invariant point is defined by three two-phase univariant equilibria. There are three divariant fields where one phase is stable.Despite its simplicity, this figure brings insights into the general behaviour.

Designation of univariant equilibria (or univariant reactions). By convention an equilib-rium curve carries the name of the phase which is not involved. Thus the equilibrium curvekyanite = sillimanite is named andalusite. It is written:

ky (and) sil

60 CHAPTER 3

Metastable extensions of univariant equilibria. A univariant equilibrium curve is the pro-jection of the intersection onto the P-T plane, of the free energy surfaces of the two assem-blages (here with only one phase) involved in the equilibrium. There is no reason that theintersection of the two planes (or surfaces) should stop at an invariant point; but beyondthis, another assemblage occurs which is more stable than the preceding ones. The univariantequilibrium curves are therefore extended beyond the invariant point in a dashed line calledthe metastable extension.

Extent of the divariant fields. It is a demonstration ad absurdum to show that the divariantfields are limited by the univariant equilibrium curves which define sectors of arc less than180°. If this was not so, the stability fields of divariant assemblages would contain themetastable extension of the univariant curves which bound the fields (Fig 38b). The divariantfields would therefore contain two domains in which the most stable assemblage would bedifferent from the stable assemblage for the sector. This situation is unacceptable.

Distribution of univariant equilibrium curves in P-T space. A univariant equilibrium curveand its metastable extension divide the P-T plane in two domains. A stable divariant assem-blage is necessarily found on one side of that line, the univariant equilibrium curves whichhave the names of the phases of that assemblage are found necessarily on the other side. In


reality, the univariant equilibria are designated by the name of the phases which do notparticipate in the reaction, it is obligatory that they are found in a domain in P-T-G space inwhich the eponymous phase is not part of the stable assemblage for the system underconsideration. This rule of metastable extension is particularly easy to understand and toapply in a system with a single independent constituent such as hasjust been examined (Fig.38a). It is so axiomatic, it performs a valuable service in constructing diagrams where n isgreater than one.

3.6.2 Construction of phase diagramsStarting with the geometrical constraints which have been detailed, and the thermodynamiccharacteristics of the three aluminosilicates, it is easy to construct the phase diagram shownin Figure 38a, without, however, being able to fix the absolute temperature and pressureconditions. The high pressure phases are those with the greatest density (smallest V); thehigh temperature phases are those which have the highest disorder in their structure (high-est S). Commonly V and S of the usual phases are available in the literature. The slopes ofthe equilibrium curves are calculated directly from the ratio .. Thus, for the reaction:

andalusite = sillimanite

The reader has the data necessary (Table 2) to construct the rest of the diagram forhimself. At this point the reactions are not calibrated in absolute values of temperature andpressure. These calibrations are effected using experimental data, principally the determi-nation of the enthalpy difference AH of the reaction by calorimetry:

At equilibrium and atmospheric pressure (normal conditions for calorimetricmeasurements):

The enthalpy differences at 1 bar for the reaction kyanite = sillimanite and kyanite =andalusite are respectively 7406 and 4351 J.mol-1. The equilibrium temperatures at atmo-spheric pressure are 599 °K and 460 °K. From these values and the slopes of the reactionsit is easy to locate the invariant point in the P-T diagram (Fig. 38a). However, in spite ofthis apparent ease there is no general agreement on the position of the triple point of thealuminosilicates in a P-T diagram.

3.6.3 Systems with two independent constituentsConsider the system Four phases are in equilibrium at the invariant point. Thecomposition of these phases must be plotted within the system, in other words on thesegment Also the choice ofphases must be realistic, and guided by observa-tions; it is possible to control, geometrically, any sort of diagram associating any sort of

62 CHAPTER 3


phases, but the diagram in this case may have no petrogenetic significance. As well, in thecase presented (Fig. 39) all the phases (quartz, andalusite, sillimanite, mullite and corun-dum) are stable under relatively low pressure and high temperature conditions, it is unreal-istic to take kyanite into consideration in place of andalusite because kyanite has never beenobserved in association with mullite. In contrast, it is possible to hesitate between an-dalusite and sillimanite to construct the invariant point (considering the uncertainty in theposition of the corresponding invariant point for kyanite-sillimanite-andalusite) and Figure39 gives two possible versions.

Four univariant equilibrium curves with three phases converge toward the invariantpoint:

and + crn (qtz) mulqtz + crn (and) muland (mul) qtz + crnand (crn) qtz + mul

By convention these equilibrium curves are written in the sense of rising temperature.The necessary indications are furnished by observation of natural assemblages in the con-tact aureoles of intrusions; at low pressure andalusite-bearing assemblages pass into mul-lite-bearing assemblages with a temperature increase. Also the corundum-quartz associa-tion has a low molar volume with respect to the other assemblages; it is from a theoreticalviewpoint the high pressure assemblage. These observations are confirmed by the valuesof molar entropies (Table 2) which allow calculation of positive values for all thesereactions. The entropy difference is therefore positive from the left going to the right.The molar volumes are calculated from the molecular and specific masses of the reactantsand products or taken directly from Table 2. as well as is calculated taking into

64 CHAPTER 3

account the stoichiometric coefficients of the products and reactants. The reaction (qtz) forexample, must be written:

All the reactions under consideration have positive slopes: Finally, therule of metastable extension will make it possible to arrange the univariant curves withrespect to one another around an invariant point (Fig. 39). The diagram outlined graphi-cally is completed giving each of the equilibrium curves its actual slope using thermody-namic data (Table 2). Ifandalusite is replaced by sillimanite, (this means that the invariantpoint is located in the stability field of sillimanite and not that ofandalusite) the figure staysthe same with a change in the slope of the univariant lines in crossing the reaction: and (ky)sil. This change is linked to the molar volume and entropy differences between the silliman-ite and andalusite bearing assemblages.

In both cases the six divariant assemblages predicted by the phase rule are separatedamong four divariant spaces limited by four univariant equilibrium curves. Each of the fourdivariant spaces has one single specific assemblage, respectively crn + qtz; and (or sil) +qtz; and (or sil) + mul; mul + qtz.

This diagram, apparently correct from the geometric and thermodynamic point ofview, presents, nevertheless, a particular problem; the rarity, read absence, in nature of thecorundum-quartz association, and the omnipresence of the association aluminosilicate (an-dalusite, sillimanite or kyanite) and quartz.

3.6.4 Systems with three independent constituentsFor two intensive variables (P,T) five phases are in equilibrium at the invariant point (Table1). Five univariant equilibria of four phases define five divariant spaces in which ten differ-ent divariant assemblages are distributed, each composed of three phases. The graphicalrepresentation ofa system with three constituents is triangular. Presenting five phases in atriangle leads to three different figures (Fig. 40):

1) The five phases define a pentagon2) Four phases define a quadrilateral with the fifth phase located inside3) Three of the phases define a triangle within which the other two are located.

The three constituent system leads to cases of collinearity when three phases arealigned on the same segment; the system is considered degenerate, and no longer has thepredicted number of univariant and divariant equilibria.

These different configurations are examined below.

3.6.4.1 Pentagonal distribution: the general case. This configuration leads to the con-struction of a nearly symmetrical form, for, by definition there are always two reactionsaround a univariant equilibrium and its metastable extension (Fig. 41). In fact all the equi-libria have two reactants and two products:

2 + 3 (1) 4 + 53 + 4 (2) 1 + 51 + 2 (3) 3 + 41 + 5 (4) 2 + 31 + 2 (5) 3 + 4

The distribution of the univariant equilibria around an invariant point obeys the rule of


metastable extension. This is automatically applied if the five phases are numbered accord-ing to the rule of diagonals (Fig.41). The five corresponding equilibria follow one anotheraround the invariant point in numerical order; or, in the absence of geologic or thermody-namic constraints it is not possible to orient the diagram in P-T space; and the numericsuccession can be established just as well in the clockwise or counterclockwise sense. Theconstruction brings out the predicted ten divariant assemblages. Each of the five divariantfields delimited by the univariant equilibria contains one unique specific assemblage (1+ 4+ 3;1+4 + 5;1+2 + 5;1 + 2 + 3; 2 + 3 + 4). This characteristic is most interesting for thedistinction between subfacies, as a specific assemblage essentially characterizes a sector ofP-T space around the invariant point, but for diverse compositions in a three componentsystem (Fig. 41).

3.6.4.2 Cases of degeneration of the system. Different cases of degeneration may be iden-tified according to the number of collinear phases.

Simple degeneration. Three phases are collinear. Consider the A'KF system and the fivephases sillimanite, muscovite, K feldspar, biotite and garnet (Fig 42a). The three firstphases lie on the A'K segment and are therefore collinear. The rule ofdiagonals applied inthis situation expunges the median phase in the collinear segment (here muscovite) fromthe triangle, without crossing any tie lines. The possible (probable?) succession of univariantequilibria around the invariant point is therefore the following:

grt + ms (Kfs) sil + btms + bt (sil) grt + Kfsms (bt) sil + Kfsbt + sil (ms) grt + Kfsms (grt) sil + Kfs

With the exception of Kfs all the equilibria are written at the onset in the direction ofa temperature increase. In fact the liberation of a vapour phase by the dehydrationreactions translates to an entropy increase (cf. the stoichiometric expression of these equi-libria in Table 3). The are therefore positive. The are also positive and thesereactions have a positive slope in the P-T diagram. There is no vapour phase in the (Kfs)equilibrium, and the balance of the A1 coordination is zero. It is impossible toappreciate the difference in entropy of the Kfs equilibrium in a simple way. This entropy

66 CHAPTER 3

difference is taken from the literature (Table 2); the highest entropy, high temperature as-semblage is sil + bt. The grt + ms assemblage has a lower molar volume than the sil + btassemblage. It is therefore the high pressure assemblage, and the slope of theequilibrium is therefore positive.

The remarkable aspect is the two univariant equilibria (bt) and (grt) correspond to thesame reaction:

ms + qtz = sil +Kfs +or ms + qtz (bt, grt) sil + Kfs +

In this particular case this reaction crosses the invariant point, it is called (grt) on oneside of the point and (bt) on the other; each of the two reactions is mixed up with themetastable extension of the other.

The consequence of the degeneration of the system is the disappearance of one divariantassemblage; nine only may be seen. But each of the divariant fields contains a specificdivariant assemblage (ms + grt + bt; sil + bt + ms; sil + bt + Kfs; sil + grt + Kfs; ms + grt+ Kfs).

As in the preceding cases (one or two independent constituents) the data from theliterature (S and V) covering the phases, as well as the stoichiometric formulation of thesereactions (Table 3) make it possible to calculate the slopes of the reaction approximately.For this specific case the exercise becomes more difficult:1) The dehydration reactions (there are four to consider) have a variable slope because of

the continuous variation of and in the fluid phase. Knowing (?) approximately

the P-T conditions of the invariant point (say 500MPa or 5 kb and 650°C) it is possibleto fix a corresponding value at these conditions for each of the two parameters (Table2). From these average values the slopes of the reactions are considered to be straightlines.

2) The ferromagnesian phases, biotite and garnet, which are involved in the reaction have

differentvalues for and as a function of their actual composition, that is their Fe/

(Fe+Mg) ratios. If the composition of the phases is known, S and V can be evaluatedusing an adjustment between the values for the iron-rich and magnesium-rich end mem-


bers. This presupposes an ideal solid solution between Fe and Mg biotite on the onehand and Fe and Mg garnet on the other (that the physical properties of these mineralspecies vary linearly between the end members). Even if this hypothesis is incorrect,it produces acceptable results at this stage of analysis.

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Double degeneration. Five phases are collinear, three and three (with one phase commonto both segments). This is the case for the A'KF system when sillimanite, muscovite,potassium feldspar, cordierite and garnet are considered. The two segments A'K and A'Feach have three phases. Application of the rule of the diagonals dictates the followingsuccession around the invariant point (Fig. 42b).

crd (ms) grt + silms (crd) sil + Kfscrd (Kfs) grt + silms + grt (sil) crd + Kfsms (grt) sil + KfsAs in the preceding case, the univariant equilibrium reactions (crd) and (grt) result

from the same muscovite breakdown reaction and the (ms) and (Kfs) equilibria result fromthe same reaction crd = grt + sil. These two reactions cross the invariant point. They haveboth been studied experimentally, which constrains the diagram (Fig 42b). Eight divariantassemblages instead of ten appear in the five divariant spaces. Each of these is character-ized by a specific assemblage (sil + grt + Kfs; sil + ms + grt; crd + ms + grt; ms + crd + Kfs;sil + crd + Kfs). A new ferromagnesian phase appears in this system: cordierite. Knowl-edge of its actual composition, as in the case of biotite and garnet, allows evaluation of thepertinent values S and V from data for pure end members (Table 2)

Regarding these latest applications, if the system is degenerate or not, it must be notedthat assemblages which contain two ferromagnesian phases (grt +bt and grt + crd) pose aparticular problem in the unequal distribution of iron and magnesium between the phases.This important aspect introduces the idea of divariant reactions which will be discussedlater on.

3.6.4.3 Other phase distribution modes; quadrilateral and triangle. The same procedureas above is applied in the two cases; geometric analysis seconded by geological observa-


tions and thermodynamic and experimental data. The numbering of the phases followingthe rule of diagonals in expelling (at the numbering time) the phase or phases containedwithin the figure, on the side where there are the least lines to cross. Respect for the rule ofmetastable extensions requires attention as the quadrilateral involves two asymmetric equi-librium reactions (one phase on one side, three on the other) and the triangle involves foursuch reactions. An example of quadrilateral construction has already been given for thesystem An example of triangular distribution in the

system is given in Figure 43.

3.6.5 The case where > + 2: petrogenetic gridsConsider the A'KF system and the six phases already used in this system: potassium feld-spar, muscovite, sillimanite, cordierite, garnet and biotite. In a three independent constitu-ent system, five phases are in equilibrium at one invariant point when two intensive vari-ables are considered. With six phases available, the following obtains:

six possibilities of five phase invariant assemblages. The univariant and divariant equilibrianumber respectively:

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Therefore an invariant point must be constructed for each of the phases under consid-eration. However, the total number of univariant and divariant assemblages is restricted(15 instead of 30; 20 instead of 60) because the six invariant points are not independent ofone another, but interconnected by the same univariant equilibria. This connection into anetwork constitutes a petrogenetic grid (Fig. 44) which allows detailed examination of theevolution of the mineral assemblages as a function of P and T. Note that:1) As each of the invariant points is characterized by the mutual stability of five phases,

one of the six phases under consideration plays no role. This phase gives its name tothe invariant point considered.

2) All the invariant points are not stable; certain, in fact, are defined by the convergenceof metastable extensions of univariant equilibrium curves (Fig. 44). This signifies thatthese points have geometric reality, but that their assemblages have no reason to exist.They are metastable from a thermodynamic point of view because there are one orseveral mineral assemblages composed of the same number of molecules of the sameindependent constituents which are more stable than these metastable assemblagesbecause they result in a lower free energy G.

An example of the geologic usage of a part of this petrogenetic grid is given in Figure 45.


3.6.6 Systems with more than three independent constituentsThe systems with more than three constituents have invariant points with (n + 2) phases inequilibrium (for two intensive variables). From these (n + 2) univariant equilibrium curves,with (n + 1) phases, are arrayed. The number of divariant assemblages with n phases growsrapidly as a function of n (Table 1). It presents a thorny problem for graphical representa-tion of all these assemblages in multidimensional space. Tetrahedral presentations (n = 4)are still usable even though they create problems of perspective, but over three dimensionsthe projections do not allow direct reading. Amongst other problems, supplementary casesof system degeneration appear, not only by collinearity (three phases on the same segment)and by coplanar array (four phases on the same plane) but also by occurring in space withmore than three dimensions which is impossible to control graphically. The present ten-dency is to explore these systems of (3 + x) constituents by matrix calculations and bydevelopment of algorithms which then construct the diagram automatically for the criticaldomain of multidimensional space.

72 CHAPTER 3

DIVARIANT REACTIONSGEOLOGIC THERMOMETRY AND BAROMETRY

For the preceding analyses, the phases involved in the reactions were always considered tobe of constant composition. However it is commonly noted:1) Ca enters the plagioclase lattice progressively (from albite, purely sodic, towards a

more calcic composition, as a function of the activity of Ca in the system).2) Al enters the amphibole lattice progressively (from tremolite and actinolite up to alu-

minous hornblende).This evolution of plagioclase and amphibole characterizes the transition between the

greenschist and the amphibolite facies on the one hand and between the albite-epidote andhornblende hornfels on the other, in relation to an increase in temperature. Further changestake place in the concentration of in biotite, also as a function of rising temperature. Itwas noted above that coexistence of ferromagnesian phases (grt - bt; grt - crd; opx - bt; opx- grt; etc.) two by two, or in a larger number, is generally characterized by an unequaldistribution of iron and magnesium between different phases. The iron and magnesiumhave comparable dimensions and fit generally in the same structural sites in ferromagnesianminerals, but their ionic radii are, however, different from one another (0.74 and 0.66Å,respectively), so that variations in the concentration in the ferromagnesianminerals generally engender significant variations in thermodynamic properties, S and Vfor example (cf. Table 2). The minimization of free energy G of an assemblage containingtwo or more ferromagnesian minerals implies, as a result, a generally unequal distributionof iron and magnesium between these phases. This behaviour results in divariant or con-tinuous reactions which are the basis for numerous methods of calculating the temperatureand pressure of crystallization.

3.7 An example of a divariant reaction: bt + sil = grt + Kfs +

Consider the reaction:(1) biotite + sillimanite + quartz = garnet + K feldspar + waterThis reaction appears as a univariant reaction in the A'KF diagram (reaction (ms) in Figure42a). Knowing that bt and grt in equilibrium and do not have the same Fe/Mg ratio, thisreaction should be looked at in an AFM projection (Fig. 46): the garnet, one of the reactionproducts, is characterized by a lower concentration than that of the biotite with whichit is in equilibrium. Its composition is, therefore, not situated on the tie line between thestarting biotite and sillimanite. From the standpoint of balance, the biotite in equilibriumwith the garnet is characterized by a higher concentration than that of the initial biotite.Examine the evolution of the reaction in a T-X diagram (Fig.46). Once conditions areattained for the reaction an infinitesimal quantity of garnetappears, characterized by an concentration clearly lower than that of the biotite of thestarting assemblage. This garnet and the new biotite (+ sil + Kfs + water) are in equilibriumfor the conditions under consideration. If the temperature rises, the composition of thegarnet and biotite in equilibrium move along the solvus of these two minerals. The amountof garnet increases and the amount of biotite decreases. At the temperature at the end ofthe reaction biotite has completely disappeared from the assemblage, and the garnet is


characterized by a concentration equal to that of the initial biotite.In fact, Figure 46 makes two univariant reactions appear, one for each “end member”.

(2) annite + sillimanite + quartz = almandine + K feldspar + water(3) phlogopite + sillimanite + quartz = pyrope + K feldspar + water

Annite and phlogopite are the ferrous and magnesian end members, respectively, ofbiotite, almandine and pyrope are the ferrous and magnesian end member garnets. All theintermediate compositions between these end members constitute a “solid solution”. Thesesolid solutions are said to be “ideal” if the physical properties vary in a linear fashion be-tween the end members; in the opposite case they are said to be “non ideal”, which is themore frequent situation. It is worthwhile examining the variation of Gibbs free energy of asolid solution, for example:

and

where is the chemical potential and the number of mols of constituent i in the phaseunder consideration.

Thus, at constant P and T, the free energy of the solid-solution varies as a function ofits composition. The lowest value of G generally corresponds to an intermediate composi-tion between the end members (Fig. 47). As a result, the intersection between the two freeenergy curves in the G-X diagram does not generally occur at the lowest value of G, butrather on a tangent to the two curves. The minimization of G is therefore realized by theequilibrium between the two phases, one relatively iron-rich and the other relatively mag-nesium-rich. The free energy of the biotite and garnet mixture is equal to the sum of freeenergies of the parts of the mixture. Figure 47 describes the different steps of the divariantreaction (1) in a G-X diagram.

Reaction (2) takes place at relatively low temperatures and reaction (3) at more el-evated temperatures. These two reactions allow construction of two petrogenetic grids,one for Fe and another for Mg (Fig. 48). All intermediate compositions on the seg-ment correspond to a reaction involving a relatively iron-rich garnet and a magnesium-richbiotite, the Fe/Mg ratio of these minerals being a function of temperature. From experi-mental and/or thermodynamic data, lines of equal value isopleths) for biotite andgarnet are traced on the P-T diagram (Fig. 48). This figure shows that the equilibriumresulting from reaction (1) is maintained over a range of nearly 250°C, in other words, overa vast divariant field. The AS of the equilibrium is high whereas the is low (cf. datafrom Table 2). The shape of reaction (2) and (3) and that of the isopleths is practicallyindependent of pressure and this divariant equilibrium is usable as a geothermometer.

As all the components of an equilibrium reaction, reactants and products, are stablealong a surface in the P-T diagram, this reaction is considered divariant. In fact the com-position of two of these phases (garnet and biotite) varies in a continual fashion as a func-tion of temperature. The isopleths demonstrate the advancement of the continuous reac-tion as a function of temperature (more and more garnet and less and less biotite) andprovides a frame for evaluating the temperature of the studied assemblages and provides a

74 CHAPTER 3

method of evaluating the temperature of the studied assemblages. These temperatures areusable if the garnet and the biotite are in equilibrium with sillimanite, potassium feldspar,quartz and water vapour, and if the concentration of water in the fluid phase is known witha reasonable precision.

3.8 Partition of iron and magnesium between biotite and garnet:An independent geothermometer

3.8.1 Principles of thermometry based on Fe-Mg exchangeA popular method of thermometry is based on the partition of Fe and Mg between coexist-ing biotite and garnet in a mineral assemblage. These two minerals effectively exchange thetwo elements when they are contiguous phases, and the exchange balance is a function oftemperature. The exchange is measured by the distribution coefficient

This exchange is described by the following reaction:

In other words: 1/3 phlogopite + 1/3 almandine = 1/3 annite + 1/3 pyropePhlogopite and annite are the magnesian and ferrous end members of biotite and al-

mandine and pyrope are the ferrous and magnesian garnet end members. If only purephases are considered at equilibrium, the chemical potentials of the two members of thereaction are equal:


But because these are solid solutions the activities (apparent concentrations) of theconstituents must be considered:

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Where is the chemical potential of a constituent i in a solid solution at the standard

state (298°K and 1 bar) and the activity of the constituent i. The activity of this constitu-ent i in the solid solution is the product of the concentration of that element and the activitycoefficient

Where α is the stoichiometric coefficient of constituent i in the solid solution. Reac-tion (2) is therefore expressed as follows:

Where K is the equilibrium constant which may be written:

Combining (4) and (6)


At equilibrium, at constant pressure, and taking into account (3), it becomes:

take from which

Therefore at constant pressure, or for small variations of molar volume, ln K varieslinearly as an inverse function of T. If the minerals involved in the reaction are ideal solidsolutions (a risky approximation which is often used) the activity coefficients are equalunits, and in consequence:

under these conditions whose value can be measured directly by chemical analysis usingthe electron microprobe, is an inverse linear function of the temperature.

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3.8.2 Experimental calibration of the reactionThe relation between ln and 1/T has been calibrated first by experimental results atpressures of 2.07 kb between 550 and 800°C (Fig 49a). The concentration of iron andmagnesium of the phases produced experimentally was determined by electron microprobe.The experimental points define a good linear relation in the diagram ln KD versus 104/T.The equation of this line was established empirically:

The temperatures calculated using this equation, on the basis of the iron and magne-sium concentration in biotite and garnet in equilibrium in rocks, are, rigorously speaking,only valid for the experimental pressures (2.07 kb). In reality, the minor difference involume involved in the reaction, in other words the small pressure influence, allows a gen-eralization of this expression over the whole stability range of the biotite-garnet associa-tion. The more general expression, proposed by Vielzeuf (1984), may also be used:

Where P is the pressure in bars and R the ideal gas constant (= 1.98726). This expres-sion takes into account the effect of pressure on the variation of free energy:

Other calibrations of this reaction have been recently proposed which are based on thesame principle, but which take into account the Ca concentration of the system as ex-pressed by the grossularite content in the garnet solid solution.

3.9 Example of a geothermobarometer:The reaction cordierite = garnet + sillimanite + quartz +

Rocks with the stable assemblage cordierite + garnet + sillimanite + quartz (± biotite) arevery widespread in metamorphic series belonging to the granulite or amphibolite facies.These constitute the kinzigitic series which sometimes crop out over several kilometres ofthickness. This mineral assemblage, therefore, remains stable over a wide range of pres-sure and temperature, and as such fulfills the definition of a divariant equilibrium. Thereaction is written:

3 cordierite = 2 garnet + 4 sillimanite + 5 quartz +(10)

This reaction only takes into consideration the ferromagnesian garnets (almandine andpyrope) to the exclusion of the calcic molecule, grossularite. The value of n which mea-sures the quantity of water involved in the reaction is open to discussion. This reactiontheoretically extends into the kyanite stability field, but kyanite-cordierite stable assem-blages are unknown in nature. The density of cordierite (2.6 to 2.7) is low compared to thatof the products of the reaction, and if water is not taken into consideration, the reactionresults in a difference in molar volume on the order of -20% (cf. data in Table 2). Thischaracteristic indicates the great barometric potential of the reaction which has been thesubject of numerous theoretical and experimental investigations. The behaviour of water


in cordierite is not quite clear, water quantities are variable, and its exact structural locationin the crystal lattice is disputed. There are, therefore, some difficulties in the thermody-namic interpretation of reaction (1), which limits its use somewhat.

3.9.1 The cordierite-garnet barometerThe cordierite and garnet which coexist in reaction (1) do not have the same Mg/Fe ratio,at equilibrium cordierite always has a greater XMg than that of garnet. Reaction (1) istherefore a divariant reaction and can be treated in exactly the same way as the biotite-garnet reaction examined above. But here, because of the low entropy and high volumedifference, pressure is the most significant parameter, and not temperature. The equilibriumis therefore examined in a PX diagram at constant temperature (Fig. 50). In this diagramthere is a vast divariant field extending over nearly 9 kb, over which the cordierite andgarnet compositions measure the advancement of the reaction as a function of pressure. Aseries of G-X diagrams analogous to those of Figure 47 (but at constant T and increasingP) would account for the evolution of the reaction (1) in the same way. Starting from thevalues of calculatedfrom experimental and thermodynamic data, thepressure of a crd-grt assemblage in equilibrium with sillimanite and quartz may be evalu-ated between 4 and 8 kb, if the crystallization temperature is known (Fig. 50).

3.9.2 The independent geothermometer crd-grtAs in the case of the association biotite-garnet, cordierite and garnet exchange iron andmagnesium as a function of the reaction:

almandine Mg-cordierite pyrope Fe-cordieriteAn analogous procedure to that which was developed for garnet-biotite equilibrium

leads to a linear relation between and the temperature (Fig 49b). The experimen-tal curve is much less well constrained than the preceding case, and values obtained fromthis relation should be viewed with caution. Several thermodynamic equations have never-theless been proposed:

where

The equations for the thermodynamic curves were established from a clearly definedexpression of and naturally care must be taken not to introduce parameters into thecalculation of that do not correspond to those that were used to define the equation.

The value may be used directly to complete the grid of Figure 50, which gives,with some errors (these may be important) the position of the assemblage crd + grt + sil +qtz in the P-T diagram, using the composition of garnet and cordierite in equilibrium.

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3.10 “Automatic” geothermobarometry

A large number of reactions with a geothermometric and geobarometric potential havebeen studied theoretically and experimentally. Some, among the most utilized, are pre-sented briefly in the appendix, either in the form of thermobarometric equations, or ingraphical form. It becomes difficult to manipulate all these reactions at the same time, anda computer programme becomes important in handling all the data. Several programmeshave been developed, which reproduce all the reactions observed within the same meta-morphic series graphically, as a function of the composition of the phases. The crystalliza-tion conditions are supposed to correspond to the best intersection of different reactioncurves. Figure 51 gives an example of this type of automatic treatment.

GEOTHERMOBAROMETRY OF FLUID INCLUSIONS

The minerals of a metamorphic rock often contain fluid inclusions, microscopic cavities(sometimes in negative crystal form) filled with a mixture of liquid, gas and even solidphases (Fig. 52a). With certain restrictions, the content of the inclusion is considered asrepresentative of the interstitial fluid phase which was present in the system at the time ofmetamorphic recrystallization, and which was trapped by the crystals during their growth.This hypothesis is only acceptable if the trapping occurred in a reservoir (mineral cavity)


both impermeable and inert, not having reacted subsequently with the imprisoned fluid.Specialists claim that quartz crystals have these qualities and their inclusions allow an effec-tive determination of the composition of the interstitial fluid present during recrystalliza-tion.

3.11 Composition of fluid inclusions

The composition of the trapped fluid phase in these fluid inclusions is obtainable by differ-ent techniques. The fluid can be extracted by crushing the samples, or by heating andanalyzing by conventional or mass spectrometry. But the techniques most commonly usedat present are microthermometry and Raman spectroscopy. In the first case, the evaluationof the inclusion’s composition is obtained by determining the freezing and melting points(appearance and disappearance of solid phases such as clathrates, the carbon dioxide eu-tectic, ice, NaCl for example) and the homogenization points (disappearance of gas phases).These data are obtained by the microscopic study of inclusions using a heating-freezingstage (-180 to 600°C) which allows direct observation of the appearance and disappear-ance of phases at measured temperatures. Raman spectroscopy furnishes an approximatechemical analysis of the inclusions (Fig 52b). These are generally composed of mixedfluids, principally composed of C - O - H - N. These fluids can be treated in a relativelysimple system containing variable quantities of NaCl in solutions

82 CHAPTER 3

and subordinate amounts of CaCl2 and KCl. Inclusions in a metamorphic series belongingto the greenschist and amphibolite facies are particularly water-rich, those of rocks recrys-tallized under granulite facies conditions, on the other hand, are water-poor and carbondioxide-rich; which explains the absence or rarity of hydrous phases in these units.

3.12 Characterization of isochores

During metamorphism the fluid phase was homogeneous at the time of trapping (or issupposed to have been homogeneous except for a few particular cases); and was monophase.During subsequent evolution toward lower temperature and pressure, the volume of theinclusion remains constant, if the effects of the compressibility of quartz are neglected, andthe overall composition does not change. The fluid evolution is, therefore, controlled by aunivariant path in P-T space, or an isochore path (volume and density constant). Theinclusion remains monophase while the P-T conditions along the isochore remain in the onefluid phase domain (Fig. 53). It begins to unmix starting at the temperature Th - homogeni-zation-unmixing temperature. When the isochore crosses or follows the univariant equilib-rium curve of the phase diagram for the appropriate composition, different gas, liquid andsolid phases appear by boiling, immiscibility and precipitation.

During this evolution the pressure in the inclusion is totally controlled by the tempera-ture, which is the only independent variable because the volume is constant (by virtue of arelationship close to that of ideal gases: PV = nRT). This evolution is therefore reversibleby simple heating of the inclusion, for example using the heating stage, and the solid andgaseous phases disappear progressively and starting at temperature the inclusion isrehomogenized and consists of a single fluid phase which has the density (or specific grav-ity) and composition of the interstitial fluid phase present at the time of metamorphism.Starting with and a series of models of varying precision, it is possible to calculate thedensity of the fluid phase, and, as a consequence, the position in P-T space of the univariantcurve, and the corresponding isochore. If all these basic assumptions are verified, the


monophase interstitial phase was necessarily on that isochore during metamorphism (Fig.53). Fluid inclusions allow the determination of thermobarometric data independent ofthose furnished by solid assemblages. They add very important information relative to theP-T conditions at the time of metamorphism and trapping.

The experimental heating of the inclusion can be raised above to no particularpurpose, the P-T conditions in the inclusion follow the isochore without any other phenom-ena being observed in the inclusion, which remains monophase. At temperature thedecrepitation temperature, the inclusion explodes because the internal pressure exceeds themechanical resistance of the inclusion walls. Effectively the microthermometric heating isapplied at atmospheric pressure, and the internal pressure of the inclusion is not compen-sated by a confining pressure as it was during metamorphism. When fluid inclusions aretrapped at high pressure the internal pressure may exceed its confining pressure during theP-T evolution of the sample; a natural decrepitation then occurs which destroys the fluidinclusion.

evaluatiion of metamorphic conditions

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